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Full text of "Analytical Mechanics"

where v is the linear velocity of the center of the hoop while the latter rotates about 0, and v1 the velocity afterwards.
DISCUSSION.—When a = 0, that is, when the two pegs coincide,
co' = co and v' = v, as they should. When a^|, co'==|,2/==|- When a = IT, co' = 0 and v' = 0, that is, the hoop comes to rest.
1.   A rod of negligible transverse dimensions and length I is moving on a smooth horizontal plane in a direction perpendicular to its length.   Show that if it strikes an obstacle at a distance a from its center it will have
an angular velocity equal to -^-, where v is its linear velocity before
meeting the obstacle.
2.   A uniform circular plate is turning about its geometrical axis on a smooth horizontal plane.   Suddenly one of the elements of its lateral surface is fixed.    Show that the angular velocity after fixing the element
equals -, where co is the angular velocity before fixing it. o
3.   A circular plate which is rotating about an element of its lateral surface is made to rotate about another element by suddenly fixing the
second and freeing the first.   Show that co' =------™------co, where co and
co' are the values of the angular velocity of the plate before and after fixing the second element, and a is the angular separation of the two elements when measured at the center of the disk.
4.   Three particles of equal mass are attached to the vertices of an equilateral triangular frame of negligible mass.   Show that if one of the vertices is fixed while the frame is rotating about an axis through the center of the triangle perpendicular to its plane the angular velocity is not changed.
6. A square plate is moving on a smooth horizontal plane with a velocity v at right angles to two of its sides. Find the velocity with which it will rotate if
(a)  one of its corners is suddenly fixed;
(b)  the middle point of one of its sides is fixed.
6.   A uniform rod of negligible transverse dimensions is rotating about a transverse axis through one end.   Find the angular uelocity with which it will rotate if the axis is suddenly removed and simultaneously a parallel axis is introduced through the center of mass of the rod.
7.   An equilateral triangular plate is rotating about an axis through